Strict solutions u of genuinely nonlinear homogeneous hyperbolic equations in two independent variables with initial data f(x) of compact support become singular after a time interval of order ∥f∥-1. In higher dimensions solutions initially of compact support are likely to have life expectancies of orders ∥f∥-2+ε at least. This is proved for the special case of solutions u(x 1,..., xn, t) of a second order equation utt = Σi,j aijuxixj, where n ≥ 3 and where the coefficients aij are C ∞-functions in the first derivatives of u, forming a symmetric positive definite matrix.